Dynamics Seminar

The dynamical Manin–Mumford conjecture predicts that a subvariety containing a dense set of preperiodic points must be special with respect to the dynamics. I will discuss a related question concerning periodic curves under regular polynomial endomorphisms of the projective plane. We show that a closed algebraic family of curves with a dense set of periodic members is preserved by an iterate of the map, and that the degree of a general member remains unchanged. I will explain the geometric ideas behind this result, including local dynamics at infinity and blow-ups. If time permits, I will discuss applications to the classification of regular polynomial endomorphisms with infinitely many periodic curves, as well as connections to a relative version of the dynamical Manin–Mumford conjecture.